It’s been a busy week… for things other than mathematics. And I never did do any mathematics or a blog post yesterday.
I did, however, do a little work on color theory last weekend and this weekend. To be specific, I have been taking photographs around my house… of things that are partly in shade. On the first day, the sky was clear and the contrast between shade and light was quite extreme. But yesterday was partly cloudy and the contrast was much less: I may not be able to get full shadow and partial shadow on the same day, but I can get both. Once I realized that, I just had to go out and take pictures.
Now, all I have for displaying the digital images is my computer monitor… and we know that that is nonlinear. Nevertheless, I can now investigate “shadow series”: the objective differences between a given color seen in shadow and seen in light.
Incidentally, I have been reminded that color perception involves significant processing by the brain, and I’ve learned a little more about it: “Through the Language Glass: Why the World Looks Different in Other Languages”, by Guy Deutscher. Much, but not all, of the book discusses color and language. He discusses some fascinating experiments. More about this down the road.
I spent a little time on control theory last weekend… I think I’m ready to do some simulations.
I have continued progressing through Dummit & Foote’s Abstract Algebra… even if I do not work every problem in the exercises, I’m trying to look at each one thoroughly.
That takes time… and while I am willing to put small amounts of time into this frequently, I don’t want to give up everything else I’m currently trying to work on. It may take me a year to work through that book. (I hope it takes no longer!)
On top of everything else, Christmas is upon us… and I have things to do. I hereby declare that I am taking a midwinter holiday, and it has begun.
Expect no technical posts before Monday, January 10. (I expect to put out some diary – happenings – posts, but we’ll see how that goes.)
When I resume, I expect that I will put out six more posts about regression… a few posts about the magic omega formula – I really want to show you what it is… and I need to pick up logic again and put out a few posts about logic and Boolean algebra.
All of these, however, are at what I call Stage II: I’m doing or have done mathematics for them, but I haven’t finished, and I haven’t isolated the mathematics for a presentation.
In addition to those multi-post topics, there are several individual posts I need to write up: the mathematics is done… I just need to “write the lecture” and convert it to a post. These are at what I call Stage III, because the specific mathematics for each post is done.
Here are the things that are that close to going out:
Constructing the final tableau for a linear programming problem, given the initial tableau and the solution (as Mathematica, for example, can provide it, using Minimize or NMinimize. The final tableau is extremely useful for sensitivity analysis… but Minimize doesn’t provide it.
Andrews curves. We encountered the term back in PCA / FA (principal component analysis, Jolliffe in particluar) but I couldn’t figure out what the actual computations were. Now I know….
Time on elliptical orbits. Distance as a function of angle is wonderful… but when will the planet or moon or spacecraft be at that angle? To say that’s “rather important” is an understatement.
I have all the calculations done for a second post on simple projectile motion.
I think I have all the calculations done for poker (five card draw, not Texas hold’em).
And all those are so close to ready to go.
In addition, there are often additional posts related to those, but I’m willing to put out what I have first.
Having gotten the final tableau, for example, I could show you how to do sensitivy analysis with it… having gotten time on an elliptical orbit (geometrically), I need to work out time on parabolic and hyperbolic orbits, and elliptical orbits, analytically. (The elliptical orbit is the only one for which I have both geometric and analytic derivations.)
There is another fascinating problem we can solve for simple projectile motion: “There it is! Shoot it!” And that problem in particular might lend itself to a second method of solution – using Clifford algebra!
And then, of course, there’s all the rest of the mathematics in the known universe….
Enough. Having done no mathematics yesterday, I am feeling a bit desperate to do some today. Probably algebra first, and then control theory.